My background: Professor of Statistics at NTNU in Trondheim, NORWAY.

My background: Professor of Statistics at NTNU in Trondheim, NORWAY.
Education: MSc in Applied Mathematics, Univ of Oslo PhD in Statistics, NTNU Work experience: Norwegian Defense Research Establishment Statoil Research interests: Spatio-temporal statistics, Computational statistics, Geoscience, petroleum applications Design of experiments, Decision analysis, value of information,

Plan for course Time Topic Thursday
Introduction and motivating examples Elementary decision analysis and the value of information Multivariate statistical modeling, dependence, graphs Value of information analysis for models with dependence Friday Examples of value of information analysis in Earth sciences Statistics for Earth sciences, spatial design of experiments Computational aspects Sequential decisions and sequential information gathering Small problem sets along the way.

Material Relevant background reading :
Eidsvik, J., Mukerji, T. and Bhattacharjya, D., Value of information in the Earth sciences, Cambridge University Press, 2015. Howard R.A. and Abbas, A.E., Foundations of decision analysis, Pearson, 2015. Many spatial statistics books: - Cressie and Wikle (2011), Chiles and Delfiner (2012), Banerjee et al. (2014), Pyrcz and Deutsch (2014), etc.

Motivating VOI examples:
Integration of spatial modeling and decision analysis. Collect data to resolve uncertainties and make informed decisions.

Motivation (a petroleum exploration example)
Gray nodes are petroleum reservoir segments where the company aims to develop profitable amounts of oil and gas. Martinelli, G., Eidsvik, J., Hauge, R., and Førland, M.D., 2011, Bayesian networks for prospect analysis in the North Sea, AAPG Bulletin, 95,

Motivation (a petroleum exploration example)
Gray nodes are petroleum reservoir segments where the company aims to develop profitable amounts of oil and gas. Drill the exploration well at this segment! The value of information is largest.

Motivation (a petroleum development example)
Reservoir predictions from post-stack seismic data! Eidsvik, J., Bhattacharjya, D. and Mukerji, T., 2008, Value of information of seismic amplitude and CSEM resistivity, Geophysics, 73, R59-R69.

Motivation (a petroleum development example)
Reservoir predictions from post-stack seismic data! Process pre-stack seismic data, or electromagnetic data?

Motivation (an oxide mining example)
Is mining profitable? Eidsvik, J. and Ellefmo, S.L., 2013, The value of information in mineral exploration within a multi-Gaussian framework, Mathematical Geosciences, 45,

Motivation (an oxide mining example)
What is the value of this additional information? Is mining profitable?

Motivation (a groundwater example)
Which recharge location is better to prevent salt water intrusion? Trainor-Guitton, W.J., Caers, J. and Mukerji, T., 2011, A methodology for establishing a data reliability measure for value of spatial information problems, Mathematical Geosciences, 43,

Motivation (a groundwater example)
Which recharge location is better to prevent salt water intrusion? Is it worthwhile to acquire electromagnetic data before making the decision about recharge?

Motivation (a hydropower example)
Adjusting water levels in 9 hydropower dams! Ødegård, H., Eidsvik, J. and Fleten, S.E., 2017, Value of information analysis of snow measurements for the scheduling of hydropower production, Energy Systems, 1-19.

Motivation (a hydropower example)
Adjusting water levels in dams! Acquire snow measurements?

Other applications Farming and forestry – how to set up surveys for improved harvesting decisions. Environmental – how monitor where pollutants are, to minimize risk or damage. Robotics - where should drone (UAV) or submarine (AUV) go to collect valuable data? Industry reliability – how to allocate sensors to ‘best’ monitor state of system? Internet of things – which sensors should be active now?

Which data are valuable?
Five Vs of big data: Volume Variety Velocity Veracity Value We must acquire and process data that has value! There is often a clear question that one aims to answer, and data should help us.

Value of information (VOI)
In many Earth science applications we consider purchasing more data before making difficult decisions under uncertainty. The value of information (VOI) is useful for quantifying the value of the data, before it is acquired and processed. ECONOMIC This pyramid of conditions - VOI is different from other information criteria (entropy, variance, prediction error, etc.)

Information gathering
Why do we gather data? To make better decisions! To answer some kind of questions! Reject or strengthen hypotheses! We will use a decision theoretic perspective, but the methods are easily adapted to other criteria or value functions (later in course).

Decision analysis (DA)
Decision analysis attempts to guide a decision maker to clarity of action in dealing with a situation where one or more decisions are to be made, typically in the face of uncertainty. Howard, R.A. and Abbas, A., 2015, Foundations of Decision Analysis, Prentice Hall.

Framing a decision situation
Rules of actional thought. (Howard and Abbas, 2015) Frame your decision situation to address the decision makers true concerns. Base decisions on maximum expected utility. ‘…systematic and repeated violations of these principles will result in inferior long-term consequences of actions and a diminishes quality of life…’ (Edwards et al., 2007, Advances in decision analysis: From foundations to applications, Cambridge University Press.)

Pirate example (For motivating decision analysis and VOI)

Pirate example Pirate example: A pirate must decide whether to dig for a treasure, or not. The treasure is absent or present (uncertainty). ? Pirate makes decision based on preferences and maximum utility or value! Digging cost. Revenues if he finds the treasure .

Mathematics of decision situation:
Alternatives Uncertainties (probability distribution) Values Maximize expected value

Pirate’s decision situation
Risk neutral!

Decision trees A way of structuring and illustrating a decision situation. Squares represent decisions Circles represent uncertainties Probabilities and values are shown by numbers. Arrows indicate the optimal decision.

Kim’s party problem Sun (0.4) Outdoor Rain (0.6) Sun (0.4) Rain (0.6)
$? Sun (0.4) Rain (0.6) Outdoor $? Sun (0.4) Rain (0.6) Porch Howard, R.A. and Abbas, A., 2015, Foundations of Decision Analysis, Prentice Hall. $? Sun (0.4) Rain (0.6) Indoors

$100 Sun (0.4) Rain (0.6) $0 Outdoor $90 Sun (0.4) Rain (0.6) $20 Porch $40 Sun (0.4) Rain (0.6) $50 Indoors

$100 Sun (0.4) Rain (0.6) $0 Outdoor $40 $90 Sun (0.4) Rain (0.6) $20 Porch $48 $40 Sun (0.4) Rain (0.6) $50 Indoors $46

Kim’s party problem   Sun (0.4) Outdoor Rain (0.6) Sun (0.4)
$100 Sun (0.4) Rain (0.6) $0 Outdoor $40 $90 Sun (0.4) Rain (0.6) $20  Porch $48  $40 Sun (0.4) Rain (0.6) $50 Indoors $46

Pirate’s decision situation

Pirate example Pirate example: A pirate must decide whether to dig for a treasure, or not. The treasure is absent or present (uncertainty). Pirate can collect data before making the decision, if the experiment is worth its price! Perfect information. Clairvoyant! Imperfect information. Detector!

Value of information (VOI)
VOI analysis is used to compare the additional value of making informed decisions with the price of the information. If the VOI exceeds the price, the decision maker should purchase the data. VOI=Posterior value – Prior value

VOI – Pirate considers clairvoyant
Conclusion: Consult clairvoyant if (s)he charges less than $1000.

PoV – decision tree, perfect information
$0 K $100 K Treasure (0.01) No treasure (0.99) Dig Don’t dig  0 K -$10 K $1 K

Pirate example - detector
Pirate example: A pirate must decide whether to dig for a treasure, or not. The treasure is absent or present (uncertainty). Pirate can collect data before making the decision, if the experiment is worth its price! Pirate makes decision based on preferences and maximum expected value! Digging cost. Revenues if he finds the treasure .

Pirate example - detector
Pirate example: A pirate must decide whether to dig for a treasure, or not. The treasure is absent or present (uncertainty). Pirate can collect data with a detector before making the decision, if this experiment is worth its price! Pirate makes decision based on preferences and maximum expected value! Digging cost. Revenues if he finds the treasure .

Detector experiment Accuracy of test:
Should the pirate pay to do a detector experiment? Does the VOI of this experiment exceed the price of the test?

Bayes rule - Detector experiment

Bayes rule - Detector experiment
Likelihood: Marginal likelihood: Posterior:

VOI – Pirate considers detector test
Conclusion: Purchase detector testing if its price is less than $460.

PoV - imperfect information
Treasure (0.16) No treasure (0.84) $100 K Dig Don’t dig $7.71 K  $7.71 K “Positive” (0.06) - $10 K $0 K Treasure (0.0005) No treasure (0.9995) $100 K $0.46 K Dig Don’t dig - $9.95 K  “Negative” (0.94) - $10 K $0 K $0 K

PV and PoV vs Digging Cost

Problem: CO2 sequestration
. . CO2 is sequestered to reduce carbon emission in the athmosphere and defer global warming. Geological sequestration involves pumping CO2 in subsurface layers, where it will remain, unless it leaks to the surface.

VOI for CO2 sequestration
. . The decision maker can proceed with CO2 injection or suspend sequestration. The latter incurs a tax of 80 monetary units. The former only has a cost of injection equal to 30 monetary units, but the injected CO2 may leak (x=1). If leakage occurs, there will be a fine of 60 monetary units (i.e. a cost of 90 in total). Decision maker is risk neutral. Data: Geophysical experiment, with binary outcome, indicating whether the formation is leaking or not. Problem: Draw the decision tree without information. Draw the decision tree with perfect information (clairvoyance). Compute the VOI of perfect information. Draw the decision tree with the geophysical experiment. Compute conditional probabilities, expected values and the VOI of geophysical data.

Value of information (VOI) - More general formulation
VOI analysis is used to compare the additional value of making informed decisions with the price of the information. If the VOI exceeds the price, the decision maker should purchase the data. VOI=Posterior value – Prior value

Risk and utility functions
Exponential and linear utility have constant risk aversion coefficient:

Certain equivalents (CE)
Utilities are mathematical. The certain equivalent is a measure of how much a situation is worth to the decision maker. (It is measured in value). What is the value of indifference? How much would the owner of a lottery be willing to sell it for?

VOI - Clairvoyance VOI=Posterior value – Prior value
Price P of experiment makes the equality. VOI=Posterior value – Prior value Assuming risk neutral decision maker!

VOI- Imperfect VOI=Posterior value – Prior value
Price of indifference. VOI=Posterior value – Prior value Assuming risk neutral decision maker!

Properties of VOI a) VOI is always positive
Data allow better, informed decisions. b) If value is in monetary units ,VOI is in monetary units. c) Data should be purchased if VOI > Price of experiment P. d) VOI of clairvoyance is an upper bound for any imperfect information gathering scheme. e) When we compare different experiments, we purchase the one with largest VOI compared with the price:

Gaussian model for profits
Gaussian, m=2, r=3 Uncertain profits of a project is Gaussian distributed.

VOI for Gaussian Uncertain project profit is Gaussian distributed.
Invest or not? The decision maker asks a clairvoyant for perfect information, if the VOI is larger than her price.

VOI for Gaussian Result:

VOI for Gaussian Gaussian cdf Gaussian pdf Result:
The analytical form facilitates computing, and eases the study of VOI properties as a function of the parameters. The more uncertain, the more valuable is information.

Problem: VOI for Gaussian
Gaussian cdf Gaussian pdf Result: Problem: Set mean to 0. Compute the VOI. Does it depend on variance? Set variance 1. Compute the VOI. Does it depend on mean? Plot VOI as a function of mean and variance.

VOI for Gaussian Gaussian cdf Gaussian pdf Result:
The analytical form facilitates computing, and eases the study of VOI properties as a function of the parameters. The more uncertain, the more valuable is information. VOI is largest for mean 0 – most undecided.

What if several projects / treasures?

What if several projects / treasures?
Where to invest? All or none? Free to choose as many as profitable? One at a time, then choose again? Where should one collect data? All or none? One only? Or two? One first, then maybe another? B

VOI and Earth sciences Alternatives are spatial, often with high flexibiliy in selection of sites, control rates, intervention, excavation opportunities, harvesting, etc. Uncertainties are spatial, with multi-variable interactions . Often both discrete and continuous. Value function is spatial, typically involving coupled features, say through differential equations. It can be defined by «physics» as well as economic attributes. Data are spatial. There are plenty opportunities for partial, total testing and a variety of tests (surveys, monitoring sensors, electromagnetic data, , etc.)

Dependence? Does it matter?
Gray nodes are petroleum reservoir segments where the company aims to develop profitable amounts of oil and gas. Martinelli, G., Eidsvik, J., Hauge, R., and Førland, M.D., 2011, Bayesian networks for prospect analysis in the North Sea, AAPG Bulletin, 95,

Dependence? Does it matter?
Gray nodes are petroleum reservoir segments where the company aims to develop profitable amounts of oil and gas. Drill the exploration well at this segment! The value of information is largest.

Joint modeling of multiple variables
C Spatial variables are often not independent! To study if dependence matter, we need to model the joint properties of uncertainties. What is the probability that variable A is 1 and, at the same time, variable B is 1 ? What is the probability that variable C is 0, and both A and B are 1 ? B

Joint pdf Discrete sample space: Probability mass function (pdf)
Continuous sample space: Probability density function (pdf)

Multivariate statistical models
The joint probability mass or density function (pdf) defines all probabilistic aspects of the distribution!

Marginal and conditional probability
Marginalization in joint pdf. Conditioning in joint pdf. Conditional mean and variance

Marginalization

Conditional probability
Venn diagram A

Conditional probability
Independence! Must hold for all outcomes and for all subsets! Unrealistic in most applications! A

Modeling by conditional probability
The joint pdf can be difficult to model directly. Instead we can build the joint pdf from conditional distributions. Holds for any ordering of variables. A

Modeling by conditionals is done by conditional statements, not joint assessment: What is likely to happen for variable K when variable L is 1? What is the probability of variable C being 1 when variables A and B are both 0? Such statements might be easier to specify, and can more easily be derived from physical principles. A

Holds for any ordering of variables. Some conditioning variables can often be skipped. Conditional independence in modeling. This simplifies modeling and interpretation! And computing! A

Conditional independence: P A C B What is the chance of success at B, when there is success at parent P? What is the chance of success at B, when there is failure at parent P? Must set up models for all nodes, using marginals for root nodes, and conditionals for all nodes with edges. A

Bayesian networks and Markov chains

Bivariate petroleum prospects example
. Conditional independence between prospect A and B, given outcome of parent! Similar network models have been used in medicine/genetics, and testing for heritable diseases.

Problem: Bivariate petroleum prospects example Problem: Compute the conditional probability at prospect A, when one knows the success or failure outcome of prospect B. Compare with marginal probability.

. Joint Failure prospect B Success prospect B Marginal probability Failure prospect A 0.85 0.05 0.9 Success prospect A 0.1 1

Bivariate petroleum prospects
. Collect seismic data :VOI - Should data be collected at both prospects, or just one of them? Partial or total? Imperfect or perfect?

. Need to frame the decision situation: Can one freely select (profitable) prospects, or must both be selected. Does value decouple? Can one do sequential selection? Need to study information gathering options: Imperfect (seismic), or perfect (well data)? Can one test both prospects, or only one (total or partial)? Can one perform sequential testing?

. Need to frame the decision situation: Can one freely select (profitable) prospects, or must both be selected. Free selection. Does value decouple? Yes, no communication between prospects. Can one do sequential selection? Non-sequential. Need to study information gathering options: Imperfect (seismic), or perfect (well data)? Study both. Can one test both prospects, or only one (total or partial)? Study both. Can one perform sequential testing? Not done here.

Bivariate prospects example - perfect
. Assume we can freely select (develop) prospects, if profitable. Total clairvoyant information

Bivariate prospects example - perfect
. Assume we can freely select (develop) prospects, if profitable. Partial clairvoyant information

Bivariate prospects example - imperfect
. Define sensitivity of seismic test (imperfect):

Bivariate prospects example - imperfect
. Assume we can freely select (develop) prospects, if profitable. Total imperfect information Can also purchase imperfect partial information i.e. about one of the prospects?

VOI for bivariate prospects example
. Partial perfect is better than imperfect total. Imperfect total better then partial perfect.

VOI for bivariate prospects example
. Price of test is 0.3

Insight in VOI – Bivariate prospects
. VOI of partial testing is always less than total testing, with same accuracy. Total imperfect test can give less VOI than a partial perfect test. Difference depends on the accuracy, prior mean for values, and correlation in spatial model. VOI is small for low costs (easy to start development) and for high cost (easy to avoid development). We do not need more data in these cases. We can make decisions right away.

Larger networks - computation
Algorithms have been developed for efficient marginalization, conditioning. Martinelli, G., Eidsvik, J., Hauge, R., and Førland, M.D., 2011, Bayesian networks for prospect analysis in the North Sea, AAPG Bulletin, 95,

VOI workflow Develop prospects separately. Shared costs for segments within one prospect. Gather information by exploration drilling. One or two wells. No opportunities for adaptive testing. Model is a Bayesian network model elicited from expert geologists in this area. VOI analysis done by exact computations for Bayesian networks (Junction tree algorithm – efficient marginalization and conditioning).

Bayesian networks, Kitchens
. Model elicited from experts. Migration from kitchens. Local failure probability of migration.

Prior marginal probabilities
. Three possible classes at all nodes: Dry Gas Oil

Prior values Development fixed cost. Infrastructure at prospect r.
Cost if dry, 0 otherwise. Revenues of oil/gas, 0 otherwise. Cost of drilling segment i.

Values . Most lucrative. But might not be most informative.

Posterior values and VOI
. Data acquired at single well.

VOI single wells . Development fixed cost.

VOI for different costs
. Development fixed cost.

VOI for different costs
. For each segment VOI starts at 0 (for small costs), grows to larger values, and decreases to 0 (for large costs). VOI is smooth for segments belonging to the same prospect. Correlation and shared costs. VOI can be multimodal as a function of cost, because the information influences neighboring segments, at which we are indifferent at other costs.

Take home from this example:
. VOI is not largest at the most lucrative prospects. VOI is largest where more data are likely to help us make better decisions. VOI also depends on whether the data gathering can influence neighboring segments – data propagate in the Bayesian network model. Compare with price? Or compare different data gathering opportunities, and provide a basis for discussion.

Never break the chain - Markov models
Markov chains are special graphs, defined by initial probabilities and transition matrices. Independence Absorbing

Markov chains (perfect information)

Avalanche decisions and sensors
Suppose that parts along a road or railroad are at risk of avalanche. - One can remove risk by cost. - If it is not removed, the repair cost depends on the unknown risk class. Data, typically putting out sensors, can help classify the risk class and hence improve the decisions made at different locations.

Avalanche decisions - risk analysis
n=50 identified locations, at risk of avalanche. At every location one can remove risk by cost 10. If it is not removed, the repair cost depends on the unknown risk class: Decision maker can secure, or not, at each location. The decisions are based on the minimization of expected costs. Prior value:

Results – different tests
All sites Only 10 first Only 11-20 Only 21-30 Only 31-40 Only 41-50 126 36 69 87 91 82 Only every second (5 measurements) gives VOI=83. Partial tests can be very valuable! Especially if they are done in interesting subsets of the domain.

Take home from this example:
. VOI varies with data gathering location Plan sensor locations wisely. VOI is largest when data are likely to help us make better decisions.

My background: Professor of Statistics at NTNU in Trondheim, NORWAY.

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